Understanding How to Find the Fourth Proportional

The question asks us to find the fourth proportional to the numbers 3, 7, and 9. This means we need to find a number, let's call it 'x', such that the ratio between the first two numbers (3 and 7) is the same as the ratio between the third and fourth numbers (9 and x).

Setting up the Proportion

We can express this relationship as a proportion: 3 is to 7 as 9 is to x. In mathematical terms, this is written as:

$3 \ratio 7 \text{ or } 3:7$

And

$9 \ratio x \text{ or } 9:x$

For these ratios to be equivalent (forming a proportion), we set them equal:

$3 : 7 :: 9 : x$

This can be written as a fraction equation:

$ \frac{3}{7} = \frac{9}{x} $

Calculating the Fourth Proportional

To find the value of 'x', we can solve this equation. A common method is cross-multiplication:

1. Multiply the first term (3) by the fourth term (x).
2. Multiply the second term (7) by the third term (9).
3. Set these products equal to each other.

Applying this to our equation:

$ 3 \times x = 7 \times 9 $

Calculate the product on the right side:

$ 3 \times x = 63 $

Now, to isolate 'x', we divide both sides of the equation by 3:

$ x = \frac{63}{3} $

Performing the division:

$ x = 21 $

Conclusion

Therefore, the fourth proportional to 3, 7, and 9 is 21. This means the ratio $3:7$ is equivalent to the ratio $9:21$. We can check this: $\frac{3}{7} \approx 0.42857$ and $\frac{9}{21} \approx 0.42857$. The ratios are indeed equal.